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<li><a class="reference internal" href="#">Model selection with Probabilistic PCA and Factor Analysis (FA)</a></li>
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  <div class="sphx-glr-download-link-note admonition note">
<p class="admonition-title">Note</p>
<p>Click <a class="reference internal" href="#sphx-glr-download-auto-examples-decomposition-plot-pca-vs-fa-model-selection-py"><span class="std std-ref">here</span></a> to download the full example code or to run this example in your browser via Binder</p>
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<div class="sphx-glr-example-title section" id="model-selection-with-probabilistic-pca-and-factor-analysis-fa">
<span id="sphx-glr-auto-examples-decomposition-plot-pca-vs-fa-model-selection-py"></span><h1>Model selection with Probabilistic PCA and Factor Analysis (FA)<a class="headerlink" href="#model-selection-with-probabilistic-pca-and-factor-analysis-fa" title="Permalink to this headline">¶</a></h1>
<p>Probabilistic PCA and Factor Analysis are probabilistic models.
The consequence is that the likelihood of new data can be used
for model selection and covariance estimation.
Here we compare PCA and FA with cross-validation on low rank data corrupted
with homoscedastic noise (noise variance
is the same for each feature) or heteroscedastic noise (noise variance
is the different for each feature). In a second step we compare the model
likelihood to the likelihoods obtained from shrinkage covariance estimators.</p>
<p>One can observe that with homoscedastic noise both FA and PCA succeed
in recovering the size of the low rank subspace. The likelihood with PCA
is higher than FA in this case. However PCA fails and overestimates
the rank when heteroscedastic noise is present. Under appropriate
circumstances the low rank models are more likely than shrinkage models.</p>
<p>The automatic estimation from
Automatic Choice of Dimensionality for PCA. NIPS 2000: 598-604
by Thomas P. Minka is also compared.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="c1"># Authors: Alexandre Gramfort</span>
<span class="c1">#          Denis A. Engemann</span>
<span class="c1"># License: BSD 3 clause</span>

<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">linalg</span>

<span class="kn">from</span> <span class="nn">sklearn.decomposition</span> <span class="kn">import</span> <span class="n">PCA</span><span class="p">,</span> <span class="n">FactorAnalysis</span>
<span class="kn">from</span> <span class="nn">sklearn.covariance</span> <span class="kn">import</span> <span class="n">ShrunkCovariance</span><span class="p">,</span> <span class="n">LedoitWolf</span>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">cross_val_score</span>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">GridSearchCV</span>

<span class="nb">print</span><span class="p">(</span><span class="vm">__doc__</span><span class="p">)</span>

<span class="c1"># #############################################################################</span>
<span class="c1"># Create the data</span>

<span class="n">n_samples</span><span class="p">,</span> <span class="n">n_features</span><span class="p">,</span> <span class="n">rank</span> <span class="o">=</span> <span class="mi">1000</span><span class="p">,</span> <span class="mi">50</span><span class="p">,</span> <span class="mi">10</span>
<span class="n">sigma</span> <span class="o">=</span> <span class="mf">1.</span>
<span class="n">rng</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">RandomState</span><span class="p">(</span><span class="mi">42</span><span class="p">)</span>
<span class="n">U</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">linalg</span><span class="o">.</span><span class="n">svd</span><span class="p">(</span><span class="n">rng</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n_features</span><span class="p">,</span> <span class="n">n_features</span><span class="p">))</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">rng</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n_samples</span><span class="p">,</span> <span class="n">rank</span><span class="p">),</span> <span class="n">U</span><span class="p">[:,</span> <span class="p">:</span><span class="n">rank</span><span class="p">]</span><span class="o">.</span><span class="n">T</span><span class="p">)</span>

<span class="c1"># Adding homoscedastic noise</span>
<span class="n">X_homo</span> <span class="o">=</span> <span class="n">X</span> <span class="o">+</span> <span class="n">sigma</span> <span class="o">*</span> <span class="n">rng</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n_samples</span><span class="p">,</span> <span class="n">n_features</span><span class="p">)</span>

<span class="c1"># Adding heteroscedastic noise</span>
<span class="n">sigmas</span> <span class="o">=</span> <span class="n">sigma</span> <span class="o">*</span> <span class="n">rng</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">n_features</span><span class="p">)</span> <span class="o">+</span> <span class="n">sigma</span> <span class="o">/</span> <span class="mf">2.</span>
<span class="n">X_hetero</span> <span class="o">=</span> <span class="n">X</span> <span class="o">+</span> <span class="n">rng</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n_samples</span><span class="p">,</span> <span class="n">n_features</span><span class="p">)</span> <span class="o">*</span> <span class="n">sigmas</span>

<span class="c1"># #############################################################################</span>
<span class="c1"># Fit the models</span>

<span class="n">n_components</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n_features</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>  <span class="c1"># options for n_components</span>


<span class="k">def</span> <span class="nf">compute_scores</span><span class="p">(</span><span class="n">X</span><span class="p">):</span>
    <span class="n">pca</span> <span class="o">=</span> <span class="n">PCA</span><span class="p">(</span><span class="n">svd_solver</span><span class="o">=</span><span class="s1">&#39;full&#39;</span><span class="p">)</span>
    <span class="n">fa</span> <span class="o">=</span> <span class="n">FactorAnalysis</span><span class="p">()</span>

    <span class="n">pca_scores</span><span class="p">,</span> <span class="n">fa_scores</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[]</span>
    <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">n_components</span><span class="p">:</span>
        <span class="n">pca</span><span class="o">.</span><span class="n">n_components</span> <span class="o">=</span> <span class="n">n</span>
        <span class="n">fa</span><span class="o">.</span><span class="n">n_components</span> <span class="o">=</span> <span class="n">n</span>
        <span class="n">pca_scores</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">cross_val_score</span><span class="p">(</span><span class="n">pca</span><span class="p">,</span> <span class="n">X</span><span class="p">)))</span>
        <span class="n">fa_scores</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">cross_val_score</span><span class="p">(</span><span class="n">fa</span><span class="p">,</span> <span class="n">X</span><span class="p">)))</span>

    <span class="k">return</span> <span class="n">pca_scores</span><span class="p">,</span> <span class="n">fa_scores</span>


<span class="k">def</span> <span class="nf">shrunk_cov_score</span><span class="p">(</span><span class="n">X</span><span class="p">):</span>
    <span class="n">shrinkages</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">logspace</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">30</span><span class="p">)</span>
    <span class="n">cv</span> <span class="o">=</span> <span class="n">GridSearchCV</span><span class="p">(</span><span class="n">ShrunkCovariance</span><span class="p">(),</span> <span class="p">{</span><span class="s1">&#39;shrinkage&#39;</span><span class="p">:</span> <span class="n">shrinkages</span><span class="p">})</span>
    <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">cross_val_score</span><span class="p">(</span><span class="n">cv</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">)</span><span class="o">.</span><span class="n">best_estimator_</span><span class="p">,</span> <span class="n">X</span><span class="p">))</span>


<span class="k">def</span> <span class="nf">lw_score</span><span class="p">(</span><span class="n">X</span><span class="p">):</span>
    <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">cross_val_score</span><span class="p">(</span><span class="n">LedoitWolf</span><span class="p">(),</span> <span class="n">X</span><span class="p">))</span>


<span class="k">for</span> <span class="n">X</span><span class="p">,</span> <span class="n">title</span> <span class="ow">in</span> <span class="p">[(</span><span class="n">X_homo</span><span class="p">,</span> <span class="s1">&#39;Homoscedastic Noise&#39;</span><span class="p">),</span>
                 <span class="p">(</span><span class="n">X_hetero</span><span class="p">,</span> <span class="s1">&#39;Heteroscedastic Noise&#39;</span><span class="p">)]:</span>
    <span class="n">pca_scores</span><span class="p">,</span> <span class="n">fa_scores</span> <span class="o">=</span> <span class="n">compute_scores</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
    <span class="n">n_components_pca</span> <span class="o">=</span> <span class="n">n_components</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">argmax</span><span class="p">(</span><span class="n">pca_scores</span><span class="p">)]</span>
    <span class="n">n_components_fa</span> <span class="o">=</span> <span class="n">n_components</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">argmax</span><span class="p">(</span><span class="n">fa_scores</span><span class="p">)]</span>

    <span class="n">pca</span> <span class="o">=</span> <span class="n">PCA</span><span class="p">(</span><span class="n">svd_solver</span><span class="o">=</span><span class="s1">&#39;full&#39;</span><span class="p">,</span> <span class="n">n_components</span><span class="o">=</span><span class="s1">&#39;mle&#39;</span><span class="p">)</span>
    <span class="n">pca</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
    <span class="n">n_components_pca_mle</span> <span class="o">=</span> <span class="n">pca</span><span class="o">.</span><span class="n">n_components_</span>

    <span class="nb">print</span><span class="p">(</span><span class="s2">&quot;best n_components by PCA CV = </span><span class="si">%d</span><span class="s2">&quot;</span> <span class="o">%</span> <span class="n">n_components_pca</span><span class="p">)</span>
    <span class="nb">print</span><span class="p">(</span><span class="s2">&quot;best n_components by FactorAnalysis CV = </span><span class="si">%d</span><span class="s2">&quot;</span> <span class="o">%</span> <span class="n">n_components_fa</span><span class="p">)</span>
    <span class="nb">print</span><span class="p">(</span><span class="s2">&quot;best n_components by PCA MLE = </span><span class="si">%d</span><span class="s2">&quot;</span> <span class="o">%</span> <span class="n">n_components_pca_mle</span><span class="p">)</span>

    <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">n_components</span><span class="p">,</span> <span class="n">pca_scores</span><span class="p">,</span> <span class="s1">&#39;b&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;PCA scores&#39;</span><span class="p">)</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">n_components</span><span class="p">,</span> <span class="n">fa_scores</span><span class="p">,</span> <span class="s1">&#39;r&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;FA scores&#39;</span><span class="p">)</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="n">rank</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;g&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;TRUTH: </span><span class="si">%d</span><span class="s1">&#39;</span> <span class="o">%</span> <span class="n">rank</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">&#39;-&#39;</span><span class="p">)</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="n">n_components_pca</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;b&#39;</span><span class="p">,</span>
                <span class="n">label</span><span class="o">=</span><span class="s1">&#39;PCA CV: </span><span class="si">%d</span><span class="s1">&#39;</span> <span class="o">%</span> <span class="n">n_components_pca</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">&#39;--&#39;</span><span class="p">)</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="n">n_components_fa</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;r&#39;</span><span class="p">,</span>
                <span class="n">label</span><span class="o">=</span><span class="s1">&#39;FactorAnalysis CV: </span><span class="si">%d</span><span class="s1">&#39;</span> <span class="o">%</span> <span class="n">n_components_fa</span><span class="p">,</span>
                <span class="n">linestyle</span><span class="o">=</span><span class="s1">&#39;--&#39;</span><span class="p">)</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="n">n_components_pca_mle</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;k&#39;</span><span class="p">,</span>
                <span class="n">label</span><span class="o">=</span><span class="s1">&#39;PCA MLE: </span><span class="si">%d</span><span class="s1">&#39;</span> <span class="o">%</span> <span class="n">n_components_pca_mle</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">&#39;--&#39;</span><span class="p">)</span>

    <span class="c1"># compare with other covariance estimators</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="n">shrunk_cov_score</span><span class="p">(</span><span class="n">X</span><span class="p">),</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;violet&#39;</span><span class="p">,</span>
                <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Shrunk Covariance MLE&#39;</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">&#39;-.&#39;</span><span class="p">)</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="n">lw_score</span><span class="p">(</span><span class="n">X</span><span class="p">),</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;orange&#39;</span><span class="p">,</span>
                <span class="n">label</span><span class="o">=</span><span class="s1">&#39;LedoitWolf MLE&#39;</span> <span class="o">%</span> <span class="n">n_components_pca_mle</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">&#39;-.&#39;</span><span class="p">)</span>

    <span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;nb of components&#39;</span><span class="p">)</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;CV scores&#39;</span><span class="p">)</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s1">&#39;lower right&#39;</span><span class="p">)</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="n">title</span><span class="p">)</span>

<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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